Periodic Coulomb Dynamics of Three Equal Negative Charges in the Field of Equal Positive Charges Fixed in Octagon Vertices

In this paper, we find an equilibrium in the Coulomb system of three equal negative point charges in the field of six equal positive point charges fixed in vertices of a convex symmetric octagon. This gives us a possibility to find periodic solutions for one-dimensional (line), two-dimensional (planar), and threedimensional (spacial) systems. Earlier, we found periodic and quasiperiodic solutions for the system of two and three negative charges in the field of two equal positive charges [1–4]. To obtain these results, we found explicit expressions for eigenvalues of the matrix U0 of second partial derivatives of the potential energy at the equilibrium of the systems. The existence of the periodic solutions followed from the Lyapunov center theorem [5–9] whenever there is no zero eigenvalue among the eigenvalues. The existence of the quasiperiodic solutions was proved with the help of the procedure of elimination of a node [8] and an application of the Lyapunov center theorem taking into account that zero eigenvalue of U0 results from a rotation invariance of the potential energy. U0 turns out to be the direct sum of two and three three-dimensional matrices for the considered here plane and spacial systems, respectively. Their simple structure permits to find their eigenvalues. U0 of the considered here space system does not possess zero eigenvalue, and the system potential energy does not have a rotation invariance. To find periodic solutions in Coulomb systems, it is not necessary to exploit the existence of their equilibria. In [10], we found the solutions for the system of n equal negative charges in the field of n equal positive charges fixed on a line (a coordinate axis). Our technique was inspired by the Siegel advanced majorant technique [11] which permits finding solutions of the Newton equation for three gravitating bodies. The Lyapunov center theorem concerns periodic solutions of the Hamiltonian systems with an equilibrium in the origin, which belong to its neighborhood, and is formulated precisely as follows.


Introduction
In this paper, we find an equilibrium in the Coulomb system of three equal negative point charges in the field of six equal positive point charges fixed in vertices of a convex symmetric octagon. This gives us a possibility to find periodic solutions for one-dimensional (line), two-dimensional (planar), and threedimensional (spacial) systems. Earlier, we found periodic and quasiperiodic solutions for the system of two and three negative charges in the field of two equal positive charges [1][2][3][4].
To obtain these results, we found explicit expressions for eigenvalues of the matrix U 0 of second partial derivatives of the potential energy at the equilibrium of the systems. The existence of the periodic solutions followed from the Lyapunov center theorem [5][6][7][8][9] whenever there is no zero eigenvalue among the eigenvalues. The existence of the quasiperiodic solutions was proved with the help of the procedure of elimination of a node [8] and an application of the Lyapunov center theorem taking into account that zero eigenvalue of U 0 results from a rotation invariance of the potential energy. U 0 turns out to be the direct sum of two and three three-dimensional matrices for the considered here plane and spacial systems, respectively. Their simple structure permits to find their eigenvalues. U 0 of the considered here space system does not possess zero eigenvalue, and the system potential energy does not have a rotation invariance.
To find periodic solutions in Coulomb systems, it is not necessary to exploit the existence of their equilibria. In [10], we found the solutions for the system of n equal negative charges in the field of n equal positive charges fixed on a line (a coordinate axis). Our technique was inspired by the Siegel advanced majorant technique [11] which permits finding solutions of the Newton equation for three gravitating bodies.
The Lyapunov center theorem concerns periodic solutions of the Hamiltonian systems with an equilibrium in the origin, which belong to its neighborhood, and is formulated precisely as follows. Theorem 1. Let an n-dimensional Hamiltonian system have real analytic Hamiltonian whose Taylor power expansion converges absolutely and uniformly at a neighborhood of the origin and begins from quadratic terms. Let also λ 1 , ⋯, λ 2n be nonzero eigenvalues of the matrix determining the linear term of the Hamiltonian vector field such that the following nonresonance relation hold for purely imaginary λ s , s = 1, ⋯, k: λ j ≠ n′λ s , s ≠ j = 1, ⋯, 2n for an arbitrary integer n′. Then, the Hamiltonian equation possesses k periodic solutions in a neighborhood of the origin such that each of them depends on a different real-valued parameter c j for some j = 1, ⋯, k. These solutions and their periods τ 1 ðc 1 Þ, ⋯, τ k ðc k Þ are real analytic functions in the parameters in a neighborhood of the origin and τ j ð0Þ = 2π/|λ j |.
The periodic solutions from this theorem take values in a neighborhood of the origin due to the fact that their expansion in the parameters c j does not contain a constant term.
The equation of motion of mechanical systems of N d -dimensional particles (bodies, charges) with masses m j and the potential energy U looks like If a potential energy possess an equilibrium x 0 j , j = 1, ⋯, N, then the potential energy with the new variables x j − x 0 j , j = 1, ⋯, N will have the equilibrium at the origin, and it is possible to apply the Lyapunov center theorem to (1).
The Coulomb potential energy U is expressed through the pair Coulomb potential. It is well known [12] that for (1) with m j = m, the eigenvalues from Theorem 1 coincide We believe that the obtained results may be useful for the semiclassical and Born-Oppenheimer approximations for quantum models of ionized molecules. The fixed positive charges and the equal positive and equal negative charges are associated with heavy nuclei and light electrons, respectively.
Our paper is organized as follows. In Sections 2, 3, and 4, we formulate our results concerning periodic solutions of the considered Coulomb equations of motion for the line, planar, and spacial systems, respectively. They are formulated in the theorems in the end of the sections.

Line Coulomb Dynamics
We consider the line dynamics of three point equal negative charges −e 0 in the field of six equal positive point charges with the same value e ′ > 0 fixed in octagon vertices with the first coordinates −a, 0, a and second coordinates ±b,± ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3a 2 + b 2 p ,±b, respectively (see the next section). The three negative point charges move along the first axis which is an invariant manifold of the planar and spacial dynamics.
The potential energy for the system is given by The equilibrium equations are given by ð∂/∂x j ÞUðx ð3Þ Þ = 0, j = 1, 2, 3:. Let us insert the equalities into it for k = 1: That is, As a result, we obtain the equilibrium relation for j = 1, 3 putting Equality ð∂/∂x 2 ÞUðx ð3Þ Þ = 0 is true since jx 0 2 − x 0 1 j = jx 0 3 − x 0 2 j. The most important information is a spectrum of the matrix of second derivatives of U. Its nondiagonal elements are easily calculated as Let U 0 j,l be this function at the equilibrium. Then, Advances in Mathematical Physics Further, Let U 0 j,j be this function at the equilibrium. Then, From the equilibrium relation, it follows that As a result, These equalities allow to represent u′ −1 U 0 j,k as simple functions of η.
Let U 0 be the matrix with the elements U 0 j,l , j, l = 1, 2, 3.
where I is unity matrix. U * 1 has identical first and third rows and this means that DetU * 1 = 0. This allows one to find roots of the characteristic polynomials p * 1 of U * 1 and p 1 ′ of U 1 ′ : Here, we subtracted the third row of −U * ðqÞ + λI from the first row. The determinant does not change after 3 Advances in Mathematical Physics the subtraction.
After that, we decomposed the determinant in the elements of the first row: The roots of p * ðqÞ are given by The roots p 1 ′ of U 1 ′ are given by Let ζ 2 ′ , ζ 3 ′ be the roots corresponding to the plus and minus before the sign of the square root, respectively: We shall always use η < 1. Let 0 < η ≤ 3 −1 , then Here, we applied 5/3 < ffiffi ffi 3 p < 7/4. This leads to ζ 2 ′ > g > v + 2 > ζ 1 ′ + 1 and there is no resonance in ζ 2 . Here, we used also Here, we applied 5/3 < ffiffi ffi We have to exclude the equality ζ 2 ′ = ζ 1 ′ with the help of Equality We proved the following proposition.
Let's prove the next proposition.
Taking into account Proposition 2, we see that ζ′ 2 > 0. Thus, we proved the proposition.
To apply the Lyapunov center theorem, we have to guarantee that ζ ′ 3 ≠ 0. Taking to the second power both terms in the expression for ζ ′ 3 , we see that this condition is satisfied if gðv − 1Þ ≠ 128. This condition is true if 0 < η ≤ 3 −1 2 since v > 9 and g > 16.
The order of charges is preserved due to the infinite repulsion and we can substitute the holomorphic functions ðx j − x k Þ −1 instead of jx j − x k j −1 in the expression for their potential energies.

Planar Coulomb Dynamics
In this section, we consider the planar system of three equal charges: −e 0 in the field of six equal positive charges e ′ fixed at the octagon vertices with the coordinates b j , 1 ≤ j ≤ 6, with the potential energy where The equilibrium is given by This gives the equilibrium relation between e 0 , e′, a, and b the same as in the previous section equating to zero the right-hand sides of these equalities for j = 1, 3 (the result is the same), taking into account x 01 1 − x 01 3 = −2a, The right-hand of (41) is zero for α = 2 and for j = 2

The equilibrium relation is given by
The second derivatives of the potential energy (39) are given by 6 Advances in Mathematical Physics Now, we shall find the equilibrium value for all the terms in these equalities. Let η, u ′ be as in the previous section. Then, we derive the following equalities: e 0 e ′ 〠 6 k=1 δ α,β e 0 e ′ 〠 6 k=1 δ α,β relying on equalities below (10) from the previous section. Let Let also T 0 j ðα, βÞ be the equilibrium value of T j ðα, βÞ. Now, we will prove the equalities with the help of the following equalities: Here, we took into account we derive from (47), (46), and (50) where Here, we took into account From the equality (48), (51), and (46), it follows where Here, we used We have also From these two equalities, one derives Now, we determine two matrices U 0 α , α = 1, 2 by the rule and renumerate indexes of coordinates in the following way: The elements of the symmetric matrices U 0 j are defined as follows: The parameters of the matrix elements are defined as follows: where v, g are the same as in the previous section. As a result, where matrix U * is defined in the previous section where we found its eigenvalues as the roots of p * ðqÞ. Now, it is not difficult to find eigenvalues of U′ 2 as roots of polynomial p′ 2 . They are given by Let ζ ′ 5 , ζ ′ 6 be the roots corresponding to the plus and minus before the sign of the square root, respectively. Then, This is true if 0 < η ≤ 1/3. In this case, ζ′ 6 < ζ′ 5 < 0. This means that there is no resonance in ζ 2 and quadratic resonance in ζ ′ 1 for the eigenvalues ζ j , 1 ≤ j ≤ 6 of U 0 . This and the Lyapunov center theorem imply the following theorem.
The last inequality follows from We proved the following theorem with the help of the Lyapunov center theorem. Theorem 6. If 0 < η ≤ 1/3 and η ≠ 1 − ð10/13Þ 2/3 , then the spacial Coulomb equation of motion (1) with m j = m, d = 3, and N = 3 and the potential energy (75) possesses the equilibrium x 01 1 = −a, x 01 2 = 0, x 01 3 = a, x 0α j = 0, j = 1, 2, 3, α = 2, 3, and two periodic solutions in its neighborhood such that each of them depends on its own real parameter c j for j = 1, 2. These solu-tions and their periods τ j ðc j Þ, j = 1, 2 are real analytic functions in a neighborhood of the origin in these parameters and τ j ð0Þ = 2π ffiffiffiffiffiffiffiffiffi m/ζ j q .

Conclusion
We have shown that the matrix U 0 of second partial derivatives of the potential energy of our spacial system at the equilibrium is the direct sum of the three dimensional matrices U 0 j , j = 1, 2, 3 such that U 0 1 and the direct sum of U 0 1 with U 0 2 coincide with matrices of the second partial derivatives of the potential energy at the equilibrium of onedimensional and planar systems, respectively. We have shown also that U 0 1 possesses two positive eigenvalues ζ 1 , ζ 2 and that ffiffiffiffi ζ j q is not in resonance with square roots of other eigenvalues for j = 1, 2 if 0 < η ≤ 1/3, η = ð5e 0 /3e ′ Þ 2/3 . The Lyapunov center theorem implies that these eigenvalues generate periodic solutions mentioned in Theorem 6 if 0 < η ≤ 1/3 and η ≠ 1 − ð10/13Þ 2/3 . The last condition guarantees that neither of the eigenvalues are zero.

Data Availability
No data were used to support this study.

Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this paper.